CSC 2402H
Methods to deal with intractability

Fall 2009

Course Outline

The course deals with systematic approaches, based on algebric and geometric reasoning to deal with intractibilitiy (aka NP-hardness).
The course will assume very little knowledge about computability and Complexity theory, and still will attempt to deliver toward its
later stages advanced concepts and state-of-the-art topics in the relevant theory, including open questions.
The course will consist of four parts.

What is a propositional Proof system, and how it is connected to P versus NP question
How standard algorithms and approximation algorithms fit into the proof complexity framework
Important Proof systems that we will consider
Resolution, Algebraic proof systems, Matrix Cut
Proof systems

Overview of lower bounds and algorithmic implications

II. Using Grobner bases to solve SAT.

Proof systems based on Hilbert's Nullstellensatz/Grobner bases.

Positive results: Grobner bases algorithms for SAT

Negative results: degree and size lower bounds

Open problems

III. (main part of the course) The convex programming approach to
approximation algorithms

What are linear programs? How do we solve Linear Programs?
We will present the simplex, ellipsoid and interior points methods.

What are semidefinite programs? solving them using ellipsoid and
interior point methods.

Using this toolbox for dealing with NP-hardness. Approximating "basic" problems. MaxCut algorithms and Sparsest cut Algorithms as prime examples. Using Linear and Semi-Definite programs to approximate MaxSAT.

The interplay between the analysis of such algorithms with combinatorial/geometrical/ probabilistic notions like concentration-of-measure, isoperimetric inequalities, expanding conditions.

Negative results: integrality gaps, LS-based integrality gaps

IV. Open problems

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Grading

There will be 4 problem sets that will make up the grade for this course.